Optimal. Leaf size=68 \[ -\frac {d^4 x (d x)^{-4+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-4+m,-n;-3+m;-\frac {b x}{a}\right )}{c^2 (4-m) \sqrt {c x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 16, 68, 66}
\begin {gather*} -\frac {d^4 x (d x)^{m-4} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m-4,-n;m-3;-\frac {b x}{a}\right )}{c^2 (4-m) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 66
Rule 68
Rubi steps
\begin {align*} \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)^n}{x^5} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {\left (d^5 x\right ) \int (d x)^{-5+m} (a+b x)^n \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {\left (d^5 x (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{-5+m} \left (1+\frac {b x}{a}\right )^n \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {d^4 x (d x)^{-4+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-4+m,-n;-3+m;-\frac {b x}{a}\right )}{c^2 (4-m) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.84 \begin {gather*} \frac {x (d x)^m (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-4+m,-n;-3+m;-\frac {b x}{a}\right )}{(-4+m) \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (d x \right )^{m} \left (b x +a \right )^{n}}{\left (c \,x^{2}\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x\right )^{m} \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x\right )}^m\,{\left (a+b\,x\right )}^n}{{\left (c\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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